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[Merged by Bors] - feat(ring_theory/polynomial/vieta): add version of prod_X_add_C_eq_sum_esymm for multiset #15008
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xroblot
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Simplify proof of multiset.prod_X_add_C_eq_sum_esymm
Clean some proofs & names Change from C + X to X + C
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It seems to me that some of the preliminary stuff was already added in other PRs. The bors d+ |
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Aug 16, 2022
…m_esymm for multiset (#15008) This is a proof of Vieta's formula for `multiset`: ```lean lemma multiset.prod_X_add_C_eq_sum_esymm (s : multiset R) : (s.map (λ r, polynomial.X + polynomial.C r)).prod = ∑ j in finset.range (s.card + 1), (polynomial.C (s.esymm j) * polynomial.X ^ (s.card - j)) ``` where ```lean def multiset.esymm (s : multiset R) (n : ℕ) : R := ((s.powerset_len n).map multiset.prod).sum ``` From this, we deduce the proof of the formula for `mv_polynomial`: ```lean lemma prod_C_add_X_eq_sum_esymm : (∏ i : σ, (polynomial.X + polynomial.C (X i)) : polynomial (mv_polynomial σ R) )= ∑ j in range (card σ + 1), (polynomial.C (esymm σ R j) * polynomial.X ^ (card σ - j)) ``` with ```lean def mv_polynomial.esymm (n : ℕ) : mv_polynomial σ R := ∑ t in powerset_len n univ, ∏ i in t, X i ``` It is better to go in this direction since there does not seem to be a natural way to prove the `multiset` version from the `mv_polynomial` version due to the difficulty to enumerate the elements of a `multiset` with a `fintype`, see this [Zulip thread](https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/.E2.9C.94.20fintype.20enumerating.20a.20multiset) and the discussion from #14908. We prove, as suggested in [#14908](#14908 (comment)) , that `mv_polynomial.esymm` is obtained by specialising `multiset.esymm`. However, we do not refactor the results of `ring_theory/polynomial/symmetric` in terms of `multiset.esymm`. From flt-regular
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Aug 20, 2022
….roots` (#14908) Specialize `multiset.prod_X_sub_C_coeff` to the root multiset of a split polynomial over an integral domain to derive the familiar Vieta's formula; update *undergrad.yaml*. Make various stylistic improvements to *polynomial/vieta.lean*: most notably, `open polynomial` to be able to omit the prefix in `polynomial.X` and `polynomial.C`. Instead, write `mv_polynomial.X` with the prefix because it's less frequent. Prove miscellaneous lemmas `list.prod_map_neg`, `multiset.prod_map_neg`, `list.map_nth_le` and `multiset.length_to_list`, which are remnants of a previous approach to prove `polynomial.vieta` superseded by #15008. See below/#14908 for the original motivation for introducing them.
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Sep 19, 2022
… version of `prod_X_sub_C_coeff` (#16424) In #15008, [`prod_X_add_C_coeff`](https://github.com/leanprover-community/mathlib/pull/15008/files#diff-08ead07a5e3b20f4db52e932b309a2ff767e486e4a0bf7d90b5520d25d95dc57L79-L81) was changed to use `multiset.esymm` in its RHS, which is defined in terms of `multiset.powerset_len` and not defeq to the original version which involves `finset.powerset_len` instead. This PR restores the `finset` version by introducing `finset.esymm_map_val`, a generalized version of `mv_polynomial.esymm_eq_multiset_esymm` (this lemma is renamed from `mv_polynomial.esymm_eq_multiset.esymm` to better conform with mathlib convention). Co-authored-by: negiizhao <[email protected]> [Zulip](https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/Help.20with.20mv_polynomial/near/297702031) Co-authored-by: Junyan Xu <[email protected]>
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… version of `prod_X_sub_C_coeff` (#16424) In #15008, [`prod_X_add_C_coeff`](https://github.com/leanprover-community/mathlib/pull/15008/files#diff-08ead07a5e3b20f4db52e932b309a2ff767e486e4a0bf7d90b5520d25d95dc57L79-L81) was changed to use `multiset.esymm` in its RHS, which is defined in terms of `multiset.powerset_len` and not defeq to the original version which involves `finset.powerset_len` instead. This PR restores the `finset` version by introducing `finset.esymm_map_val`, a generalized version of `mv_polynomial.esymm_eq_multiset_esymm` (this lemma is renamed from `mv_polynomial.esymm_eq_multiset.esymm` to better conform with mathlib convention). Co-authored-by: negiizhao <[email protected]> [Zulip](https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/Help.20with.20mv_polynomial/near/297702031) Co-authored-by: Junyan Xu <[email protected]>
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Sep 21, 2022
… version of `prod_X_sub_C_coeff` (#16424) In #15008, [`prod_X_add_C_coeff`](https://github.com/leanprover-community/mathlib/pull/15008/files#diff-08ead07a5e3b20f4db52e932b309a2ff767e486e4a0bf7d90b5520d25d95dc57L79-L81) was changed to use `multiset.esymm` in its RHS, which is defined in terms of `multiset.powerset_len` and not defeq to the original version which involves `finset.powerset_len` instead. This PR restores the `finset` version by introducing `finset.esymm_map_val`, a generalized version of `mv_polynomial.esymm_eq_multiset_esymm` (this lemma is renamed from `mv_polynomial.esymm_eq_multiset.esymm` to better conform with mathlib convention). Co-authored-by: negiizhao <[email protected]> [Zulip](https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/Help.20with.20mv_polynomial/near/297702031) Co-authored-by: Junyan Xu <[email protected]>
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This is a proof of Vieta's formula for
multiset:where
From this, we deduce the proof of the formula for
mv_polynomial:with
It is better to go in this direction since there does not seem to be a natural way to prove the
multisetversion from themv_polynomialversion due to the difficulty to enumerate the elements of amultisetwith afintype, see this Zulip thread and the discussion from #14908.We prove, as suggested in #14908 , that
mv_polynomial.esymmis obtained by specialisingmultiset.esymm. However, we do not refactor the results ofring_theory/polynomial/symmetricin terms ofmultiset.esymm.From flt-regular